How can skew-symmetric matrices be proven to be a subspace of M_{n \times n}(F)?

[RavenCpu]

Hello all! I finally decided to join this forum after quite a long time of lurking haha. Anywho, to the point. A friend of mine is taking linear algebra and is having a lot of issues with the below posted problem. I'm trying to help her with it, but sadly I'm having issues with it as well haha. I would appreciate any help you all could offer me in doing this proof. I've gotten it all solved up until I have to prove that M_{n\times n}(F) = W_1 \oplus W_2. So basically, I'm on the last step.

Prove:
A matrix M is called a skew-symmetric if M^t = -M. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set W_1 of all skew-symmetric n x n matrices with entries from F is a Subspace of M_{n \times n}(F). Now assume that F is not of characteristic 2, and let W_2 be the subspace of M_{n \times n}(F) conisting of all symmetric n x n matrices. Prove that M_{n\times n}(F) = W_1 \oplus W_2

 

[AKG]

Rewriting it so it's legible:

Hello all! I finally decided to join this forum after quite a long time of lurking haha. Anywho, to the point. A friend of mine is taking linear algebra and is having a lot of issues with the below posted problem. I'm trying to help her with it, but sadly I'm having issues with it as well haha. I would appreciate any help you all could offer me in doing this proof. I've gotten it all solved up until I have to prove that M_{n\times n}(F) = W_1 \oplus W_2. So basically, I'm on the last step.

Prove:
A matrix M is called a skew-symmetric if M^t = -M. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set W_1 of all skew-symmetric n x n matrices with entries from F is a Subspace of M_{n \times n}(F). Now assume that F is not of characteristic 2, and let W_2 be the subspace of M_{n \times n}(F) conisting of all symmetric n x n matrices. Prove that M_{n\times n}(F) = W_1 \oplus W_2

 

[AKG]

1. Prove that W_1,\, W_2 are subspaces
2. Prove W_1 \cap W_2 = \{ 0\}
3'. Find bases \beta _1,\, \beta _2 and show that \beta _1 \cup \beta _2 is a basis for M_{n\times n}(F). This is easy since it's just a matter of comparing |\beta _1| + |\beta _2| to dim(M_{n x n}(F)).
3''. Instead of 3', you could do the following: given any square matrix A, use your bases to compute the projection of A onto each of those subspaces, and show that the sum of the two projections is A.
3'''. You should do 3' or 3'', but another way is to prove that for any given matrix A, the equation A = (1/2)(A + A^t) + (1/2)(A - A^t) holds, and that a) A + A^t is symmetric, b) A - A^t is skew symmetric, and c) (1/2) makes sense because F is not of characteristic 2. This is the most efficient way to prove it, but it requires you to know how to decompose A in the first place. Doing the first two ways will actually teach you how, in the future, you can go about proving that some vector space is the direct sum of some subspaces. This third approach is just a handy tid-bit of knowledge, not a very instructive approach though.

 

[RavenCpu]

Thanks for rewriting it :-) I'm not too familiar with symbols and the sort in forums. It looks so clean now!

Thank you for the help! I'll give it an attempt and see what comes of it.